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systems using QPSK modulation
Petros Ramantanis
To cite this version:
Petros Ramantanis. Contribution to the analysis of optical transmission systems using QPSK mod-ulation. Economics and Finance. Institut National des Télécommunications, 2011. English. �NNT : 2011TELE0020�. �tel-00765380�
Thèse n° 2011TELE0020
Thèse présentée pour l’obtention du diplôme de
DOCTEUR DE TELECOM & MANAGEMENT SUDPARIS
Doctorat délivré conjointement par
TELECOM & Management SudParis et l’Université Pierre et Marie Curie - Paris 6
Spécialité : Electronique et communications
Par Petros Ramantanis
Contribution à l’étude des systèmes de
transmission optique utilisant le format de
modulation QPSK
Soutenue le 30/09/2011 devant le jury composé de :
Président : Georges Alquié Rapporteur : Alberto Bononi Rapporteur : Philippe Emplit Examinateur : Yann Frignac
Examinateur : Jean-Christophe Antona Directeur de thèse : Badr-Eddine Benkelfat
Thesis n° 2011TELE0020
PhD Thesis
TELECOM & MANAGEMENT SUDPARIS
PhD jointly delivered by
TELECOM & Management SudParis and Université Pierre et Marie Curie - Paris 6
Specialty: Electronics and communications
Petros Ramantanis
Contribution to the analysis of optical
transmission systems using QPSK modulation
Date of defense: 30/09/2011. Jury composition:
Head of the jury: Georges Alquié Reviewer: Alberto Bononi
Reviewer: Philippe Emplit Jury member : Yann Frignac
Jury member: Jean-Christophe Antona Thesis director: Badr-Eddine Benkelfat
A simple “thanks” for the people that have contributed directly or indirectly to this manuscript is certainly not enough. Nevertheless, the few words that follow give me the chance to remember and thank them with all my heart, not only because of their help, but most importantly, for all the priceless moments that we have shared. First and foremost, I would like to express my infinite gratitude to my thesis advisor, office colleague and dearest friend, Yann Frignac. His qualities of character, his technical expertise, as well as his end-less patience and energy, have brought into our everyday office life an atmosphere of scientific stimulation and creativity, only to be (vio-lently) interrupted by hearty laughs. A few of the strongest moments include the deliriums in front of our velleda whiteboards, the con-ference submissions made just a few minutes before the deadline, as well as the philosophical discussions at the end of the day, in which we were usually rebuilding the world from scratch (alias “refaire le monde”).
At this point I would also like to warmly thank my thesis director, Badr-Eddine Benkelfat. Without his kindness, his continuous encour-agement and support at all levels, this work would have been impos-sible. When I think of Badr-Eddine I realize that I could simply not wish for a better or friendlier thesis director.
Looking back in time, I owe a great thanks to my dear friend Stelios Sygletos, who has also been my diploma thesis tutor in the National Technical University of Athens, back in 2006. Apart from sharing with me his scientific expertise, Stelios was also a constant source of inspiration and support during the years that followed. Sometimes I
this amazing field.
Remembering the years that followed, I would also like to thank my professors Fran¸cois Hache, Jean-Michel Jonathan and Nathalie West-brook, not only for their teaching skills but also for their careful ad-vice and help during my master’s year, which was also my first year in France. Furthermore, I would also like to thank S´ebastien Bigo, who has kindly offered me an internship in Alcatel-Lucent back in 2007 and who was also responsible for leading me towards my PhD at Telecom SudParis. In the same context, I also owe a great thanks to Jean-Christophe Antona, my internship tutor and a member of my thesis jury, for the various inspirational discussions that we had at the time and during the years that followed, as well as for his helpful remarks on my thesis manuscript.
Concerning my actual PhD work, I would also like to thank Gabriel Charlet, not only for leading the COHDEQ40 project - the main frame of my thesis, but also for his sharp technical suggestions, as well as his constant help and support throughout these years. Moreover, I would also like to warmly thank all my other Alcatel-Lucent colleagues with which I have interacted on technical issues, Emanuel Seve, Massimil-iano Salsi, Thierry Zami, Clemens Koebele and Edouard Grellier, as their suggestions and comments helped me orientate my thoughts and reinforce my technical background. Next to that, I also owe a special thanks to our two Italian colleagues from the Parma university that I have also met in Alcatel-Lucent: my dear friend (and Linux geek fellow), Paolo Serena, for our rich technical interaction, the inspir-ing conversations and the open source simulation program Optilux, as well as Alberto Bononi for accepting to participate in my thesis defense jury, offering at the same time his serious and well-supported feedback on my manuscript.
my supervision. Apart from her important scientific contribution, I have also appreciated her character and personality, which made working with her a great pleasure. I would also like to thank my dear friend and PhD colleague Djalal Bendimerad, who has started working in Telecom SudParis nearly at the same time as me, as well as my other PhD fellows Jordi Vuong and Aida Seck with all whom I feel extremely lucky to have shared over the past years (and still share) a wonderful interaction from both a scientific and a human point view. Furthermore, I also owe a great thanks to the computer experts of Telecom SudParis Eric Doutreleau, Jehan Procaccia and Franck Gillet for bearing my continuous complaints and offering me their help in crucial moments, as well as to my friend Lazaros In-epologlou for always managing to give me well-adapted solutions to a number of programming questions. Finally, I would like to thank my foosball pals Tony, Javier, Anis and Aymen for turning our coffee breaks into special and memorable events.
Ending up, I would like to thank my parents, relatives and friends for their continuous moral support throughout all these years, while, last but not least, I would like to thank my girlfriend Elli, to whom I dedi-cate this manuscript, for all her caring and understanding throughout this particularly hard period of our lives.
Acronyms 1
1 Introduction 4
2 Theoretical framework 9
2.1 Concepts of Digital communications . . . 9
2.1.1 Introduction . . . 10
2.1.2 Signal representation . . . 11
2.1.3 Digital modulation . . . 12
2.1.3.1 Amplitude Shift Keying modulation . . . 13
2.1.3.2 Phase Shift Keying modulation . . . 15
2.1.3.3 Differential encoding . . . 16
2.1.3.4 Spectral characteristics of modulated signals . . . 17
2.1.4 Systems impacted by Additive White Gaussian Noise . . . 19
2.1.4.1 Additive White Gaussian Noise and Signal-to-Noise ratio . . . 20
2.1.4.2 Signal statistics and Bit Error Probability in On-Off Keying . . . 22
2.1.4.3 Signal statistics and bit error probability in Phase Shift Keying . . . 25
2.2 Lightwave communication systems . . . 48
2.2.1 Optical fibers . . . 49
2.2.1.1 Attenuation . . . 50
2.2.1.2 Chromatic dispersion . . . 52
2.2.1.4 Fiber Non-linearities . . . 61
2.2.1.5 Nonlinear Schr¨odinger equation . . . 64
2.2.2 Transmitters, signal modulation and modulation formats . 66 2.2.2.1 General characteristics . . . 67
2.2.2.2 Generation of Amplitude Shift Keying modulation 68 2.2.2.3 Generation of Binary Phase Shift Keying modu-lation . . . 70
2.2.2.4 Generation of Quaternary Phase Shift Keying mod-ulation . . . 70
2.2.3 Signal Reception . . . 71
2.2.3.1 Photodiodes . . . 72
2.2.3.2 Demodulation of Amplitude Shift Keying . . . . 73
2.2.3.3 Demodulation of Differential Phase Shift Keying 74 2.2.4 Amplifiers and noise . . . 75
2.2.5 Coherent detection in lightwave communications . . . 78
2.2.5.1 Principle of the coherent detection . . . 80
2.2.5.2 Phase diversity homodyne receiver . . . 82
2.2.6 Inter-channel nonlinear impairments . . . 85
2.2.7 Intra-channel nonlinear impairments . . . 87
2.2.8 Fundamental limitations in fiber optic channels . . . 89
2.2.9 Impairment mitigation . . . 92
2.2.10 Numerical simulations . . . 94
2.2.10.1 Introduction . . . 94
2.2.10.2 The Split Step Fourier Method . . . 95
2.2.10.3 Resources used for our numerical simulations . . 96
3 Investigation of M-ary sequences: application on the perfor-mance evaluation of QPSK transmission systems 98 3.1 Introduction : pseudo-random sequences . . . 98
3.2 Finite Fields : a short review . . . 102
3.2.1 Introduction . . . 102
3.2.2 Prime Finite Fields and polynomials . . . 103
3.3 Pseudo-random sequences: generation methods and properties . . 110
3.3.1 Generation of pseudo-random sequences: a method based on shift-registers . . . 110
3.3.2 Properties of pseudo-random sequences . . . 114
3.3.2.1 The autocorrelation function property . . . 115
3.3.2.2 The window property and de Bruijn sequences . . 119
3.4 Non pseudo-random sequences . . . 123
3.5 Performance assessment of dispersion-managed links by different sequence types . . . 127
4 Propagation influence on the statistics of QPSK modulated sig-nals in single-channel dispersion managed systems 133 4.1 Introduction . . . 133
4.2 Motivations . . . 134
4.3 Simulations setup and examples . . . 136
4.4 Statistical measures . . . 143
4.5 Criterion of cumulative nonlinear phase . . . 144
4.6 Dispersion Management Optimization and constellation shape . . 150
4.6.1 Optimization of dispersion management for phase and am-plitude . . . 150
4.6.2 Global phase shift . . . 158
4.7 The constellation shape based on the data sequences carried by the signal . . . 165
4.7.1 Pattern-dependent nonlinear degradation . . . 165
4.7.2 Most degraded subsequences . . . 169
4.8 Conclusion . . . 174
5 Conclusion 176
A Differential encoding and decoding 179
ASE Amplified Spontaneous Emission
ASK Amplitude Shift Keying
AWGN Additive White Gaussian Noise
BEP Bit Error Probability
BER Bit Error Rate
BPSK Binary Phase Shift Keying
CMA Constant Modulus Algorithm
CPE Carrier Phase Estimation
CW Continuous Wave
DBPSK Differential Binary Phase Shift Keying
DCF Dispersion Compensating Fibers
DD Direct Detection
DGD Differential Group Delay
DM Dispersion Management
DMGD Differential Mode Group Delay
DMPSK Differential M-ary Phase Shift Keying
DPSK Differential Phase Shift Keying
DQPSK Differential Quaternary Phase Shift Keying
DSP Digital Signal Processing
EDFA Erbium-Doped Fiber Amplifier
FEC Forward-Error Correction
FFT Fast Fourier Transform
FIR Finite Impulse Response
FWM Four Wave Mixing
HM Half Mirror
i-FWM Intra-channel Four Wave Mixing
IM Intensity Modulation
ISI Inter-Symbol Interference
i-SPM Intra-channel Self Phase Modulation
i-XPM Intra-channel Cross Phase Modulation
LO Local Oscillator
MAP Maximum A Posteriori
MPSK M-ary Phase Shift Keying
MZM Mach-Zehnder Modulator
NLSE Non-Linear Schr¨odinger Equation
NRZ Non Return to Zero
OOK On-Off Keying
OSNR Optical Signal to Noise Ratio
PBS Polarization Beam Splitter
PDF Probability Density Function
PDM Polarization-Division Multiplexing
PLL Phase Lock Loop
PMD Polarization Mode Dispersion
PR Pseudo-Random
PRBS Pseudo-Random Binary Sequence
PRQS Pseudo-Random Quaternary Sequence
PRS Pseudo-Random Sequence
PSBT Phase Shaped Binary Transmission
PSD Power Spectrum Density
PSK Phase Shift Keying
QAM Quadrature Amplitude Modulation
QPSK Quaternary Phase Shift Keying
QWP Quaternary Wave Plate
RZ Return to Zero
SNR Signal to Noise Ratio
SPM Self Phase Modulation
WDM Wavelength-Division Multiplexing
XPM Cross Phase Modulation
Introduction
”...Where is the Life we have lost in living?
Where is the wisdom we have lost in knowledge?
Where is the knowledge we have lost in information?...”
T. S. Eliot, The Rock (1934)
Communications and information technologies are ubiquitous in modern soci-eties. Talking over mobile phones, exchanging photos and videos over the Internet and researching online information, is currently a commonplace for most people. To go even further, it is often said that communications and information tech-nologies are also radically changing the idea or form of society itself[44]. It cannot
be doubted that a major part of this revolution came as a result of the dramatic increase in transmitted information capacity brought by optical fibers[5].
The cornerstone for the development of fiber-optic transmission systems was doubtless the advent of single-mode, low-loss, optical fibers[64],[65],[39],[83]. This
ac-complishment, accompanied by other critical advances in the domains of laser technology, nonlinear optics, etc., has given birth to the emerging domain of lightwave communications, roughly since the beginning of the 80’s. During the following years, further technological innovations have increased even more the dynamics of the field. A milestone in this direction has been the development
sion of the installed information capacity with the birth of Wavelength-Division Multiplexing (WDM) systems. Moreover, the optical amplification itself has led to the possibility of counteracting the fiber attenuation at the cost of added noise, thus offering longer propagation distances without the need of signal regeneration. In parallel, continuous technological advances in all sub-domains of lightwave communications were continuously pushing up capacity from the order of some Gb/s during the 80’s, up to tens of T b/s in the past decade, following an almost exponential growth[46]1. Apart from WDM, in this “quest for capacity growth”, several other technological solutions contributed in a decisive manner, such as the use of Forward-Error Correction (FEC) that can considerably increase the system noise tolerance, Raman Amplification that can increase the Signal to Noise Ratio (SNR) compared to EDFAs or Dispersion Management (DM) that increases system tolerance against nonlinear effects. Nevertheless, it cannot be doubted that a radical change in the field of optical communications came along with the appearance of a modern coherent detection implementation, based on fast electronics instead of an optical Phase Lock Loop (PLL)[107]. The main
advantage of coherent receivers, in general, is the possibility to use complex mod-ulation schemes that can offer increased spectral efficiencies. Already some years before the appearance of coherent detection, the principle of modulation and di-rect demodulation of Quaternary Phase Shift Keying (QPSK) (which is maybe the most typical example of a spectrally efficient modulation format) has been demonstrated in optical communication systems by the authors of[55]. However, yet another advantage of the electronics-based coherent detection was the poten-tial use of a flexible, software-based and cost-effective signal equalization at the received end, based on a programmable Digital Signal Processing (DSP) unit. This has given the opportunity to easily take advantage of Polarization-Division
1Nevertheless, in the last few years, several investigations have been focusing on a forthcom-ing “capacity crunch” (i.e. demand for installed capacity overcomforthcom-ing offer), as a consequence of approaching to the fundamental capacity limits of single-mode fibers[34],[36]. One of the
possi-ble solutions to continue the increase of system capacity is the use of multi-mode or multi-core fibers.
achieved in the electronic domain. Furthermore, many linear impairments such as chromatic dispersion or Polarization Mode Dispersion (PMD) were also shown to be compensated in a cost-effective and flexible way, in the electronic domain.
In the context of this thesis which began in 2008, we have focused on the problem of terrestrial long-haul or ultra-long haul communications, i.e. the prob-lem of setting up a transmission link where the involved distances have an order of magnitude of about 200 up to 900 km (long-haul) or 1000 up to 5000 km (ul-tra long haul), where the propagated distance between two amplifiers is usually about 100 km[40].
Furthermore, we have adopted an axis that was primarily based on two of the aforementioned techniques, i.e. dispersion management and coherent detection, investigating the nonlinear degradation of both signal quadratures (i.e. phase and amplitude), instead of just amplitude, that was the case for most investigations concerning On-Off Keying (OOK) modulation. While linear degradation caused by chromatic dispersion can be easily compensated by a simple use of Dispersion Compensating Fiberss (DCFs), the technique of dispersion management consists of wisely distributing DCFs throughout the line, in order to reduce the degrada-tion caused by nonlinear effects. However, the possibility of coherent receivers to electronically compensate dispersion together with nonlinearity issues, encour-aged tin some cases he abandoning of dispersion management schemes with in-line dispersion compensation. Nevertheless, in many terrestrial networks, dispersion management comes as a legacy of existing fiber infrastructure that we wish to “upgrade”, by a simple adjustment of terminal devices. Alternatively, one may consider dispersion management as a technique that may be used in addition or in parallel to coherent receiver algorithms, as an extra force against nonlineari-ties. As a result, throughout this manuscript we have adopted a vision where in all cases we consider systems with a variable dispersion management that may also benefit from a coherent receiver and a DSP unit.
More precisely, based on numerical simulations of principally single-channel
toOOK-modulated signals in the same context (2) understand the possible influ-ence of adapted correction algorithms as a function of the different degradation patterns resulting from different dispersion management schemes.
In chapter §2 we present general, introductory notions, of digital communi-cations and classical fiber optics. As the extended use of digital communicommuni-cations concepts into the domain of fiber-optics is relatively new (for example multi-level modulation is relatively new to fiber optic communications, since the domain was almost entirely dominated by OOK/direct detection schemes), we attempt to clarify the theoretical basis of the two domains.
In chapter §3 we address the problem of using multi-level Pseudo-Random Sequences (PRSs) in numerical simulations instead of adapted versions of Pseudo-Random Binary Sequences (PRBSs), when dealing with multi-level modulation formats. Since PRSs present a particular interest in various other areas of telecommunications, their generation process and properties have been thor-oughly studied since the 1950s. However, to the best of the author’s knowledge, the information concerning the generation of multi-levelPRSs as well as the exact definition of their properties, is quite scattered within the existing bibliography. For reasons of clarity, we first systematically review all the underlying theory be-hind these special sequences, based on the theory of Finite Fields. Furthermore, since non-PRSs are very commonly used for numerical simulations or laboratory experiments, we also present simple numerical tools that can be used to char-acterize these non-PRSs with respect to their “pseudo-random characteristics”. Finally, we present numerical simulation results that support the necessity of using Pseudo-Random Quaternary Sequences (PRQSs) in the context of QPSK
modulation, revealing in parallel, the special system configurations for which the usage of PRQSs is most critical.
Finally, the section §4is dedicated to the investigation of the nonlinear prop-agation of QPSK-modulated signals, in the context of 40 Gb/s, single-channel transmission, with a variable dispersion management. Our first goal was to ver-ify the validity of laws, developed in the context of OOK modulated signals,
all cases we have decoupled the transmission from the reception and our numeri-cal results were mostly based on the complex signal statistics (of both amplitude and phase), instead of a simple Bit Error Rate (BER). Furthermore, we have thoroughly investigated the variation of these statistics for a variable dispersion management, as well as the relative variation or correlation between the degra-dation of the amplitude and phase quadratures, that results in constellations of very different shapes. Finally, since the degradation of signals in a single-channel transmission comes in part from Inter-Symbol Interference (ISI), we have em-ployed a phenomenological analysis, focusing this time on the statistics of the complex samples of isolated symbols, grouped with respect to the data carried by their neighboring symbols. This approach elucidates the mechanism of ISI in the context of QPSK-modulation with dispersion management and reveals the possibilities of an adapted algorithm that can compensate for signal distortion.
Theoretical framework
Everything should be as simple as it is, but not simpler.
Albert Einstein
In this chapter we review the fundamental theoretical background used through-out this manuscript. For reasons of clarity, the chapter is divided in two sections. In the first section, we review some fundamental mathematical concepts of digital communications that apply to all communication systems, without any reference to how these concepts are implemented in optics. Such concepts, such as complex modulation formats, have been widely used in the past in a context of wireless communications, but they have been only recently introduced in the fiber optics domain. In the second section we review the physics and the special characteris-tics of the optical fiber communication channel.
2.1
Concepts of Digital communications
The field of digital communications has been rapidly developing throughout the last century, principally driven by the explosion of communication networks. At the basis of all communication scenarios, lies the capacity of transmitting in-formation between the different entities of a communication network, principally using two modes of communication: transmitting information from one point (i.e. node of a network, user, entity etc) to multiple other points, often referred to as
broadcasting, or alternatively transmitting information from one point to another (i.e. from one user to another or from one network node to another etc), often referred to as point-to-point communication. In what follows we are exclusively focusing on a point-to-point communication context.
2.1.1
Introduction
Point-to-point digital communication systems are often schematically represented using the structured (or layered) view of figure 2.1, with arrows representing the flow of information. The message to be transmitted, is initially converted into a bit sequence by the input transducer. Next, this sequence is further transformed into a new bit sequence by the source encoder that reduces the number of bits compressing the data, while the channel encoder increases the number of bits, adding a redundancy that allows for an error correction at the receiver side. Finally, the modulator maps the bit stream into waveforms and the waveforms are transmitted into the channel. At the receiver end, the inverse procedure is followed, with waveforms being initially mapped back into bits, and bits passing from the channel decoder, the source decoder and finally the output transducer, eventually recovering the transmitted sequence, if the passage from the channel was error free.
Input transducer Output transducer Source encoder Source decoder Channel encoder Channel decoder Digital modulator Digital demodulator Info! bits bits
bits! less bits
Tx
bits! more bits bits! waveformsRx
Channel
Figure 2.1: Coherent detection principle
With the structured representation we achieve the decomposition of the com-munication problem into independent, or semi-independent sub-problems. In this
manuscript we only refer to problems associated with the last two layers, i.e. the problem of digital modulation/demodulation and the problem of signal transmis-sion in the channel, that is the fiber-optic channel in our case, supposing in all cases that the bit sequence {Ibn} enters the digital modulator layer.
Depending on the choice of the modulation format, the bit sequence {Ibn} may by alternatively represented as a symbol sequence {Isn}, by considering blocks of k = log2M bits and substituting each block with its decimal equivalent, i.e. decimal numbers in the range {0, 1, ..., M − 1}. Then, the symbol sequence {Isn} is transformed into the waveform sequence {In} by mapping each symbol into one of the M possible waveforms.
As it is the case in most communication channels, the single-mode fiber-optic channel exhibits a limited bandwidth of about 0.4 µm (or 50 T Hz) around the telecommunication wavelength 1.55 µm (or 193 T Hz)1. As a consequence, the waveforms chosen to represent the data are in most cases band-limited, modulated signals, i.e. signals with a limited frequency extent, centered around the carrier frequency, with the carrier frequency residing inside the channel bandwidth.
2.1.2
Signal representation
Let g(t) denote a bandpass signal with a central frequency fc and a Fourier transform G(f ) = Z +∞ −∞ g(t)e−j2πftdt (2.1) or inversely g(t) = Z +∞ −∞ G(f )ej2πf tdf (2.2)
The signal g(t) is linked to its complex envelope, i.e. its low-pass equivalent ˜
g(t), by the relation
g(t) = Reg(t)e˜ j2πfct (2.3)
1This bandwidth is mainly fixed by the wavelength window where the absorption of step-index silica fibers is minimal as it will be discussed in section§2.2.1.1.
, or equivalently in the Fourier domain G(f ) = 1 2 h ˜ G(f − fc) + ˜G∗(−f − fc) i (2.4) ˜
g(t) is generally a complex function described as ˜
g(t) = x(t) + jy(t) = a(t) exp (jθ(t)) (2.5)
and a(t), θ(t), x(t), y(y) are real functions of t.
Since the modulated signal can be sufficiently described it terms of its complex envelope and the modulation frequency, in the following, we may just refer to the complex envelope ˜g(t) of the signal.
2.1.3
Digital modulation
Suppose that we want to transmit the symbol sequence Isn =Is1, Is2, ..., IsLseq
. Mapping the symbols of Isninto digital waveforms, yields the information bearing sequence In=I1, I2, ..., ILseq
. The baseband equivalent of the overall waveform entering the channel ˜g(t) can be expressed as a sum of partial baseband wave-forms ˜gn(t), n = 1, 2..., Lseq, one for each of the Lseq transmitted symbols[89], often
referred to as pulses, or mathematically
˜ g(t) = Lseq X n=1 ˜ gn(t) (2.6)
where ˜gn(t) is the pulse corresponding to the nth transmitted symbol. Going further, ˜gn(t) can be written as
˜
gn(t) = Inp(t − nT ) (2.7)
, where R = 1/T is the symbol rate and p(t) is often referred to as the shape-forming pulse, whose choice is generally a part of the system design and optimization. Nevertheless, for reasons of simplicity, in what follows, we consider that p(t) is the rectangular or gate function, i.e. p(t) = rect(t/T )1, defined as
1Even in practice, p(t) is not very different from rect(t). Since rect(t) is not practically realizable we are looking for a pulse shape that is practically realizable and not very different from the ‘ideal” for of rect(t).
rect(t) = 1, −12 ≤ t ≤ 1 2 0, elsewhere (2.8)
Having chosen an appropriate pulshaping function, mapping from the se-quence {Isn} to the sequence {In} is the problem of choosing a modulation format. In the following we present the two families of modulation formats, used throughout this manuscript, i.e. Amplitude Shift Keying (ASK) and Phase Shift Keying (PSK). In both cases, the complex envelope of each transmitted symbol can be written in the form
˜
gn(t) = Am· p(t), m = 0, 1, ..., M − 1 (2.9)
, where Am is defined in the following paragraphs.
2.1.3.1 Amplitude Shift Keying modulation
In ASKmodulation, we map each transmitted symbol into a discrete amplitude level, i.e. in equation (2.9), each Am corresponds to a discrete real number. In other words, the waveform of each symbol is a constant signal with an amplitude level defined by Am. We also usually refer to the different Am as theASKstates. The simplest way to chose Am is to set
Am = m · A, m = 0, 1, ..., M − 1 (2.10)
,where A is a fixed difference between amplitudes with a consecutive m. Set-ting M = 2 to eq. (2.10), we get a set of two amplitude levels, {0, A}, resulting in
the simplest and most widely used modulation format in optical communications, the 2-ASKor more simply On-Off Keying (OOK). In effect, with this modulation format we transmit a signal for the symbol “1” or nothing for the symbol “0”, as shown in figure 2.2a.
In a different scenario, amplitude levels can be chosen so that they are sym-metric with respect to zero, i.e.
Am = (2m + 1 − M)A
"0"
"1"
A
0
(a) Constellation forOOK
modulation.
"0"
"1"
2
A
!
2
A
(b) Constellation for symmetric 2-ASK mod-ulation."0"
"1"
2
2
A
!
2
2
A
(c) Constellation for symmetric 2-ASK modulation with same average power asOOK.
Figure 2.2: Comparison of different 2-ASK scenarios.
In this case, a signal with a non-zero amplitude is transmitted for all possible symbols. Setting in eq. (2.11) M = 2 results in a “symmetric” 2-ASK, with a constellation shown in figure 2.2b.
It is important to note that if the two symbols (”0” or “1”) have an equal probability of appearance, the average signal power1 in the first scenario is A2
2 , while in the second scenario it is A2
4 , while the distance between the two symbols is kept constant and equal to A. “Normalizing” the second scenario to the average power of the first, results in a signal with a distance of√2A between the two levels, as shown in figure 2.2c. It is obvious that since the two symbols are separated by a greater distance between the two levels, a greater tolerance is achieved against degradation effects like noise. This tolerance is going to be quantified in a following section for the simple degradation form of additive white Gaussian noise.
We should also note that, in practice, the mapping from symbols to amplitude levels is slightly different than the one of equations(2.10)and (2.11). In fact, the mapping may chosen in such a way that neighboring symbols differ to only one bit, so that mistaking a symbol for one of its neighboring ones will only yield a single bit error. This mapping is called Gray encoding[89].
1As it is going to be discussed in following sections, the average signal power is an indicator of the penalties due to nonlinear degradations: in general, higher average signal power yields a higher penalty due to nonlinearities. Therefore, it is very common to compare modulation formats in the common basis of an equal average power[45].
4
!
3
4
!
3
4
!
"
4
!
"
(a) Constellation for QPSKmodulation.
"0"
"1"
!
0
(b) Constellation forBPSK mod-ulation.
Figure 2.3: Constellations of PSK modulation formats.
In conclusion, amplitude modulation consists of mapping the symbol sequence Isn = {Is1, Is2, ...}, where Isn ∈ {0, 1, ..., M − 1}, into the waveform sequence In= {I1, I2, ...}, where In = AIsnp(t).
2.1.3.2 Phase Shift Keying modulation
In PSK modulation, we map each symbol into a discrete phase level, i.e. in equation (2.9), each Am gets a discrete imaginary value, with one discrete phase level out of the M such possible levels, i.e.
Am = ej·(m
2π
M+θ0), m = 0, 1, ..., M − 1 (2.12)
, where θ0 is a fixed phase offset. PSK modulation with phase levels (or M discrete states) is also often referred to as M-ary Phase Shift Keying (MPSK).
Equivalently, the symbol sequence Is = {Is1, Is2, ...}, where Isn∈ {0, 1, ..., M − 1} is mapped to the waveform sequence I = {I1, I2, ...}, where In = Am· p(t).
Similar to ASK, Gray encoding is also used in PSK to ensure that adjacent phase levels are mapped from symbols that differ to only 1 bit. A constellation example for 4-PSK, (also widely known as QPSK), with Gray encoding is shown in figure 2.3a. A constellation example for 2-PSK or BPSK, is shown in figure
format. In the context of this manuscript we will primarily focus on the QPSK
modulation format.
2.1.3.3 Differential encoding
For reasons that will become clear in section §2.1.4, in many cases, instead of coding information on an absolute amplitude or phase level, information is coded in the amplitude or phase difference between two adjacent symbols. In this case, we refer to differential modulation or modulation with memory, since the sequence symbols are correlated. In effect, the initial information sequence is transformed into a new sequence, often referred to as (differentially) pre-coded sequence.
XOR 1!symbol! delay Input Sequence {Ibn} Pre-coded Sequence {Pbn}
Figure 2.4: Generation of pre-coding sequence in DBPSK
In DBPSK for example, given the initial binary sequence {Ibn}, the
genera-tion of the pre-coded sequence {P bn} is achieved by the circuit shown in figure
2.4. Alternatively, the pre-coding operation may be simply described by the equation[102]
P bn= Ibn⊕ P bn−1 (2.13)
, where the symbol ⊕ represents an addition modulo 2 (i.e. a port XOR). In other words, if the new symbol to be transmitted is identical to the one transmit-ted in the previous slot we transmit a “0”, whereas if the symbol is different we transmit a “1”. In Differential Quaternary Phase Shift Keying (DQPSK), pre-coding can be achieved in the exact same way, representing quaternary symbols as couples of bits, i.e. {00, 01, 10, 11} and performing the modulo 2 addition bitwise. On the other hand, supposing that the sequence arriving at the receiver side is the sequence {Rbn}, in order to recover the initially transmitted sequence (if no
errors occurred during transmission) the same XOR operator must be emulated between adjacent symbols, i.e.
Dbn = Rbn⊕ Rbn−1 (2.14)
In optical communications however, the pre-coding operation for DQPSK is slightly more complex1. The input binary sequence {Ib
n} is first split in two sequences {IPk} and Qk of half the length of {Ibn}. For example, we may assign to {IPk} the odd bits of {Ibn}, i.e. {IPk} = {Ib2k−1}, and to Qk the even bits of {Ibn}, i.e.{Qk} = {Ib2k} , where k = 1, 2, .... Each couple (IPk, Qk) represents a quaternary symbol of the new quaternary sequence {Isk}. Pre-coding consists of converting the sequence Isk = (IPk, Qk) into a new sequence P sk = (Uk, Vk), where[55]
Uk= (IPk⊕ Qk) · (IPk⊕ Uk−1) + (IPk⊕ Qk) · (Qk⊕ Vk−1) Vk = (IPk⊕ Qk) · (IPk⊕ Vk−1) + (IPk⊕ Qk) · (Qk⊕ Uk−1)
(2.15) We need to underline that since pre-coding maps the initial sequence {Ibn} into a new sequence {P bn}, it also, in general, modifies the statistics of the in-formation sequence {Ibn}. Furthermore, as the initial sequence is modified, the detection (demodulation) of differentially encoded signals in optical communica-tions, involves special circuits that ensure the recovering of the initial information sequence, as it is going to be detailed in section §2.2.3.3.
2.1.3.4 Spectral characteristics of modulated signals
Since the modulated signal is transmitted in a band-pass channel, it is funda-mental to study its spectral characteristics. As we can note from equation (2.7), the waveform sequence I is based on the random input symbol sequence Is and therefore the resulting modulated signal is a stochastic process.
Consider now that g(t) in equation (2.3) is a sample function of the overall stochastic process. With the additional assumptions that g(t) is a wide-sense
sta-1We describe the pre-coding operation that was introduced in optical communications by the authors of[55]. The demodulation, also performed in optics, in shown in the same paper.
tionary processes with a zero mean1, it can be shown[89] that the autocorrelation
functions of the modulated and band-pass signal are linked with the relation φgg(τ ) = Reφ˜g˜g(τ )ej2πfcτ
(2.16) , or equivalently, their Power Spectrum Density (PSD) with the relation
Φgg(f ) = 1
2[Φ˜g˜g(f − fc) + Φ˜g˜g(−f − fc)] (2.17) As it is evident from the above, the PSD of the modulated signal can be uniquely determined by the PSD Φ˜g˜g(f ) of the low-pass signal ˜g(t). As ˜g(t) is a function of the information sequence In, in order to go further we need to make an assumption over the statistics of the information sequence Isn and therefore, over the waveform sequence I. More precisely we suppose that the information sequence In is a wide-sense stationary process2, with a mean value
µi = E [In] (2.18)
and an autocorrelation function φii(m) = 1
2E [I ∗
nIm+n] (2.19)
The autocorrelation function of ˜g(t) is defined as φ˜g˜g(t + τ, t) =
1 2E [˜g
∗(t)˜g(t + τ )] (2.20)
and using equations (2.19), (2.6) and (2.7) it can be shown that ˜g(t) is a cyclostationary process with power spectral density
¯ Φ˜g˜g(f ) = 1 TΦii(f ) |P (f)| 2 (2.21) , where thePSD Φii of the information sequence {In} is defined as
1A wide-sense stationary stochastic process X(t) has a fixed mean value (independent of t) and an autocorrelation function E [X(t1), X(t2)] = ϕ (t1, t2) = ϕ (t1− t2) = ϕ (τ ).
2i.e. the mean value is time-independent and for the autocorrelation function holds ϕ(t1, t2) = ϕ(t1− t2) = ϕ(τ )[56]. A detailed discussion and a demonstration in the context of pseudo-random is going to be shown in chapter §3.
! "
0 2 N t#
( ) nn t$
t(a) [Autocorrelation function of white noise. 0 2 N ( ) nn f ! f (b)PSD.
Figure 2.5: Characteristics ofAWGN.
Φii(f ) = +∞ X k=−∞
ϕii(k) · e−2πfkT (2.22)
and P (f ) is the Fourier transform of the pulse-shaping function p(t).
As it can be easily seen from equation(2.21)that ¯Φ˜g˜g(f ) depends on both the pulse-shaping function p(t) and the correlation characteristics of the information sequence, expressed via the term Φii.
2.1.4
Systems impacted by Additive White Gaussian Noise
The term “noise” is used to designate spontaneous fluctuations of the quantity used to transfer the information in our transmission system. As it is going to be discussed in §2.2, the two major sources of signal distortion in optical communi-cation systems are noise and fiber nonlinearities. The combined result of these distortions is often referred to in the literature as “non-linear phase noise”1.
Focusing on noise, however, it may be shown that the dominant source of noise is spontaneous emission noise, added by amplifiers (discussed with more details in §2.2.4), a noise source that can be adequately described by the notion of AWGN as shown in [50].
1Linear effects like chromatic dispersion also introduce a signal distortion, but when they act alone they are easily compensated. By nonlinearities we refer to the interplay between linear and nonlinear effects.
In the following sections, we describe the properties of AWGN and we focus on the degradation purely coming from anAWGN. Being primarily interested in the two modulation formats reviewed in section §2.1.3 (i.e. ASK and PSK), in the following sections we derive the statistics of the signal cylindrical quadratures (i.e. amplitude and phase) in the presence of AWGN. More precisely, we derive their corresponding Probability Density Functions (PDFs), we calculate their first moments and we, finally, derive the bit error probabilities for ASK or PSK
modulation.
2.1.4.1 Additive White Gaussian Noise and Signal-to-Noise ratio
Denoting by n(t) the stochastic AWGNprocess with a zero mean, n(t) is defined via its autocorrelation function (figure 2.5a)
ϕnn(τ ) = N0
2 δ (τ ) (2.23)
, or equivalently by it power spectral density (figure2.5b)
Φnn(f ) = N0
2 (2.24)
, where N0 is given in W/Hz. We note that AWGN has a flat spectrum
density for all frequencies and an autocorrelation function corresponding to a Dirac function. These characteristics are linked to completely random processes. In chapter§3 we will see that pseudo-random sequences have properties that are very similar to the properties of AWGN, even though they are sequences created in a deterministic way.
Considering that anAWGNis applied to an ideal low pass filter of bandwidth W , the resulting stochastic process N (t) will be characterized by a new couple of
PSD and autocorrelation, given by
ΦN N(f ) = N0
2 , −W < f < W
0, otherwise (2.25)
The variance σ2
N of the noise samples, being equal to the expected value of the noise power (since the process mean is zero) is given by the relation
σN2 = EN2 = ϕN N(0) = N0W (2.27)
When a (low-pass) signal ˜g(t) is impacted by an AWGN n(t), the resulting signal ˜r(t) may be written as
r(t) = ˜g(t) + n(t) (2.28)
, where n(t) = x(t) + j · y(t) is the complex sample function of the AWGN
process with its two components x(t) and y(t), i.e. the in-phase and the quadra-ture noise component correspondingly, are jointly Gaussian real random variables
AWGNprocesses with the characteristics of each being described by the equations
(2.23), (2.24).
In such a case, denoting with P = A2 the signal average power, the Signal to Noise Ratio (SNR), (denoted as ρ in what follows), is defined as
ρ = A 2 2σ2 = P 2σ2 = P W N0 (2.29) Let X and Y be the identically distributed random variables of the sample functions x(t) and y(t). More precisely,
pX(x) = √1 2πσe
−x2
2σ2 (2.30)
Setting A = X2 the PDF of A reads:
pA(a) = √ 1
2πaσe − a
2σ2 (2.31)
The characteristic function of A is given by:
ψA(iu) = E
eiuA = f {pA(a)} = √ 1
1 − i2uσ2 (2.32)
Defining a new random variable that describes the optical power of the signal S with S = R2 = X2+ Y2 and as X and Y are identically distributed, we get
ψS(iu) = 1
and
pS(s) = e −2σ2s
2σ2 (2.34)
The distribution of the equation(2.34)is also known as chi-square distribution with two degrees of freedom. This distribution is very useful in optical communi-cations, since the photocurrent is directly proportional to the optical power (see section§2.2.3.1). Therefore, the received photo-current after a cascade of additive white Gaussian noise sources, is going to follow a chi-square distribution.
Finally, defining the complex modulus R = √S and using (2.34) we get the
pdf of R:
pR(r) = r · e −r2
2σ2
σ2 (2.35)
On the other hand, by defining Θ = tan−1 Y X
and noting that X and Y are identically distributed, we get that Θ is uniformly distributed in [0, 2π], or
pΘ(θ) = 1
2π (2.36)
An alternative way of calculating the PDFs of R and Θ is to consider the joint distribution of X and Y , i.e. pX,Y(x, y) = 2πσ1 e−x2+y22σ2 , if we perform a
variable change with R = √X2+ Y2 and Θ = tan−1 Y X
we finally get that pR,Θ(r, θ) = r
2πσ2e− r2
2σ2 and consequently we get the equations (2.35) and (2.36).
2.1.4.2 Signal statistics and Bit Error Probability in On-Off Keying
In OOK modulation, as presented in §2.1.3.1, the possible symbols are mapped into two distinct amplitude levels, named x0 and x1. The bit error probability in this case is given by
BEP = p(0) · P (1/0) + p(1) · P (0/1) (2.37)
, where p(0), p(1) are the probabilities of initially transmitting a “0” or a “1” and P (1/0), P (0/1) are the probabilities of falsely detecting a “1” and falsely detecting a “0” correspondingly. If “0”s and “1”s are transmitted with the same probability, equation (2.37) becomes
BEP = 1 2(P (1/0) + P (0/1)) (2.38) signal ti m e
x
0x
Dx
1 s ig n a l probability P(1/0) P(0/1) Figure 2.6: Q factorAssuming an AWGN, the PDF of level “1” can be described by the relation
p1(x) = √ 1
2πσ1e
−(x−x1)2σ2 2
1 (2.39)
, while thePDF of level “0” is described by the relation
p0(x) = √ 1
2πσ0e
−(x−x0)2σ2 2
0 (2.40)
, where σ0 and σ1 are the standard deviations of level “0” and level “1” correspondingly.
Supposing that the decision threshold is set to xD as shown in figure 2.6 and using equations (2.39) and (2.40), the error probabilities P (1/0) and P (0/1) are calculated as
P (1/0) = +∞ Z xD p0(x)dx = 1 2erf c xD− x0 √ 2σ0 (2.41) and P (0/1) = xD Z −∞ p1(x)dx = 1 2erf c x1− xD √ 2σ1 (2.42) Finally, combining the equations(2.38), (2.41)and (2.42), the bit error prob-ability reads BEP (xD) = 1 4 erf c x1− xD √ 2σ1 + erf c xD− x0 √ 2σ0 (2.43) As it is evident from the equation (2.43), the Bit Error Probability (BEP) depends on the threshold xD and it is minimized when BEP′(xD) = 0. Solving the equation BEP′(xD) = 0 for the optimal threshold xD and supposing that lnσ1
σ0 ≈ 0 weget
xD = σ0x1+ σ1x0
σ0+ σ1 (2.44)
Defining the Q factor as
(xD− x0) σ0 = (x1− xD) σ1 ∆ = Q (2.45)
and substituting(2.44) into (2.45)we finally get
Q = x1− x0
σ0+ σ1 (2.46)
Q is directly proportional to the separation of the levels averages x1 − x0, while it is inversely proportional to the sum of their noise standard deviations σ0 and σ1. At the same time, we intuitively expect that the bit error rate will be high when the levels are close to each other, or/and when the noise variance in each level increases.
In effect, by setting the threshold to its optimal value of eq. (2.44), or equiv-alently replacing(2.45) into(2.43), we get an expression linking bijectively the Q factor to the bit error probability1
BEP = 1 2erf c Q √ 2 (2.47) In conclusion, with the above procedure, we can analytically link the BEP
of an OOK transmission system corrupted byAWGN to the Q factor. We have
to underline that the above analysis is accurate only in the context of OOK
modulation and when the system is impacted only by AWGN. In optical com-munications, as it will be briefly discussed in§2.2.3.1, photo-current noise may be assumed to have Gaussian statistics and therefore, this analysis is quite accurate
for back-to-back measurements, employing OOK modulation. However, when a
transmission line is present, the amplified spontaneous emission noise generated by the amplifiers overwhelms the shot or thermal noise and, as noted before, the optical power and therefore the photocurrent, follow a chi-square distribution.
Nevertheless, since the Q factor is linked to the BEP bijectively, for com-parison reasons, in many cases we convert measured bit error probabilities for arbitrary modulation formats, into a “fake” Q factor by taking the inverse func-tion of (2.47), i.e.
Q =√2 · erfc−1(2 · BEP ) (2.48)
In the rest of this manuscript, when we refer to a Q factor we implicitly mean the “fake” Q factor defined by the equation (2.48).
2.1.4.3 Signal statistics and bit error probability in Phase Shift
Key-ing
Without loss of generality we may consider that the equivalent low pass signal corresponding to one state of a PSK signal, in a system impacted by AWGN is
1A distinction has to be made between the Q function, commonly used in the digital com-munications literature, and the Q factor. The Q function, is defined as Q(x) = 12erf c(√x
2) and it may be used to simplify the expressions of bit error rates. It may also be used to link the
BEPto the Q factor but this is not done here for reasons of clarity. As a consequence, in this manuscript we are always using the letter “Q” to refer to the Q factor.
described by the relation1
S = A + N (2.49)
Figure 2.7: CW + noise
, where A is a real fixed number and N represents theAWGN along the two signal quadratures with N = N1+ j · N2, N1 and N2 being identically distributed, Gaussian Random variables with zero mean and standard deviation σ (figure2.7). The received signal vector may then be represented by a new complex random variable: S = A + N1+ jN2 = X + jY , with PDFs of the real and imaginary part given by pX(x) = √1 2πσe −(x−A)22σ2 (2.50) pY(y) = 1 √ 2πσe −2σ2y2 (2.51)
, and a joint PDF of X and Y given by: pX,Y(x, y) =
1 2πσ2e
−(x−A)2+y22σ2 (2.52)
Passing to polar coordinates by defining
R = √X2+ Y2 (2.53)
and
Θ = tan−1Y
X (2.54)
the joint pdf is now given by
pR,Θ(r, θ) = r 2πσ2e
−r2+A2−2Ar cos θ2σ2 (2.55) Integrating(2.55) we can get the PDFs of R and Θ, i.e.
pR(r) = r σ2e −r2+A22σ2 Io Ar σ2 , r ≥ 0 (2.56)
, where I0 is the modified Bessel function of the first kind, and pΘ(θ) = e−ρ 2π + √ρ · e−ρ·sin2θ √ 4π cos (θ) erf c(− √ ρ · cos θ) (2.57)
, where the term ρ appearing in (2.57) is the SNR, defined in (2.29) with P = A2. For high SNRs, equation (2.57) reads
pΘ(θ) ≈ r ρ πe −ρ·sin2θ cos (θ) (2.58)
Moreover, it can be easily noted from eq. (2.57) that pΘ(θ) is a function of cos(θ) it is periodic and as a periodic function it can be expanded in a Taylor series. Following this expansion eq. (2.57) can be re-written as[18]
pΘ(θ) = 1 2π + e−ρ2 2 √ρ √ π ∞ X m=1 (−1)m m2 h Im−1 2 ρ 2 + Im+1 2 ρ 2 i cos (mθ) (2.59)
The last expression is very useful for the calculation of the error rates, as it will be discussed later in the chapter.
Finally, as mentioned before, since the photocurrent is directly proportional to the optical power of the signal, it is interesting to investigate the statistics of
the optical power. Setting S = R2 with S being the random variable describing the optical power, the pdf of S reads
pS(s) = e −s+A2 2σ2 I0(A √s σ2 ) 2σ2 (2.60)
with a characteristic function
ψS(ju) = E[ejuS] = e
juA2 1−j2uσ2
1 − j2uσ2 (2.61)
As shown from equation(2.57), pΘ(θ) can be effectively expressed as a function of only θ and ρ. In figure 2.8 we plot pΘ(θ) for various values of ρ.
Figure 2.8: pΘ(θ) for various values of ρ.
2.1.4.3.1 Calculation of the first moments
Having the exact expression for the PDF pX(x) of the random variable X, one may calculate the m-order moments of X using the formula
E[Xm] = +∞ Z −∞
xmpX(x)dx (2.62)
E[(X − µX)m] = +∞ Z −∞
(x − µX)mpX(x)dx (2.63)
, where µX is the average value of the random variable X with µX= E[X]. The 2ndorder central moment, or variance, noted as σ2
X, is of particular inter-est as it provides a rough measure of the random variable “spread”. However, we are usually based on the variance square root, or standard deviation. Developing
(2.63) for m = 2, we get
σX =
q
E [X2] − µ2
X (2.64)
Having the exact expressions for the PDFs of R and Θ we can use the equa-tions (2.62) and (2.64) in order to calculate the standard deviations σR and σΘ. Beginning with the variable R, we get its average
E[R] = µR = σ r π 2e −ρ2 h (1 + ρ) · I0 ρ 2 + ρ · I1 ρ 2 i (2.65) , the 2nd order moment
E[R2] = A2+ 2σ2 = 2σ2(1 + ρ) (2.66)
and the standard deviation σR= σ r 2 (1 + ρ) − π2e−ρh(1 + ρ) · I0ρ 2 + ρ · I1 ρ 2 i2 (2.67) In order ot get more intuition on equation (2.67), in figure 2.9a we plot the quantity σR
σ against ρ (in linear scale). We note that the function σR/σ has a lower bound σR σ ρ=0 = p 2 −π 2 ≈ 0.6551, while σR σ < 1 with limρ→∞ σR σ = 1. In
a realistic optical-communication systems, SNRs typically1 take values around ρ = 10, for which σR is slightly lower but very close to σ.
1In optical communications, instead of SNR, it is common to use the Optical Signal to Noise Ratio (OSNR) in a reference bandwidth of 12.5 GHz or 0.1 nm around 1.55 µm (OSNR
will be properly defined in section §2.2.4). For reasons of comparison we just note that for a noise bandwidth of 50 GHz (for example, almost entirely containing a 20 Gbaud signal) ρ = 10 corresponds to OSN R0.1 nm= 13 dB.
0 2 4 6 10 20 0.7 0.8 0.9 1 ρ σ R /σ
(a) σR/σ as a function of the signal-to-noise ratio in a system impacted by
AWGN. 0 5 10 15 20 0 0.5 1 1.5 2 ρ σ Θ (rad)
(b) σΘ as a function of the signal-to-noise ratio in a system impacted by
AWGN.
Figure 2.9: σΘ and σR/σ as a function of ρ. On the other hand, for the variable Θ we get its average value
E[Θ] = 0 (2.68)
, its variance (using eq. (2.59))
E[Θ2] = π 2 3 + 2e −ρ2√πρ +∞ X m=1 (−1)m m2 h Im−1 2 ρ 2 + Im+1 2 ρ 2 i (2.69)
and its standard deviation[19]
σΘ = v u u tπ2 3 + 2e −ρ2√πρ +∞ X m=1 (−1)m m2 h Im−1 2 ρ 2 + Im+1 2 ρ 2 i (2.70)
Moreover, a simplified relation can be obtained for high SNR (ρ ≥ 10) by
using the PDF of the equation (2.58),or
σΘ ≈
r 1
2ρ (2.71)
In figure2.9b, we plot σΘ as a function of ρ. Considering the limits of σΘ we can easily see that σθ(0) =
q π2
3 ≈ 1.8138 and limρ→∞σΘ = 0.
Fusing on the approximate expression (2.71), we see that σΘ is bijectively linked to ρ As standard deviations provide an estimation for the spread of a random variable, it serves as a simple indicator of the signal degradation, in cases
where we are not interested in performing a strict calculation of the symbol error probability or run a brute-force Monte-Carlo Bit Error Rate (BER) estimation. In coherently detected PSKmodulation, as the decision is based on the phase of the signal, σΘ provides the appropriate quantity that captures the phase degradation and therefore, the phase standard deviation may provide a rough estimation of the PSKsignal degradation. However, it should be underlined that the detection method plays an important role and it may drastically change the statistics of the detected signal quadratures1.
A careful observation of the equations (2.70) and (2.67) reveals that σΘ is a function of just ρ, while σR is a function of both the noise standard deviation σ and ρ. Suppose now that un unknown noise source corrupts our signal. Since we have no precise information about this source, it is very useful, at a preliminary stage, to measure its resemblance to AWGN, by means of simple measurements of the complex mean Aest, the standard deviations σΘ, σR and an estimation of the “noise” standard deviation σest. For this, we introduce at this point the parameter Bn=∆ σΘ,estσest σR,esth|Aest|2 2σ2 est (2.72)
, where the function h(ρ) is analytically calculated using the equations(2.70)
and (2.67), defined as h(ρ) = s π2 3 + 2e− ρ 2√πρ +∞P m=1 (−1)m m2 h Im−1 2 ρ 2 + Im+1 2 ρ 2 i q 2 (1 + ρ) −π 2e−ρ (1 + ρ) · I0 ρ2 + ρ · I1 ρ2 2 (2.73)
From the above definition, it is obvious that when the signal deformation is caused by AWGN, the distortion is symmetric, yielding Bn = 1. On the other hand, when Bn > 1, the phase quadrature has a larger variance compared to the variance of the amplitude quadrature, while when Bn < 1, the amplitude quadrature has a larger variance compared to the phase quadrature variance.
1For example, in contrast to coherently detected systems, differentially systems do not yield the same statistics, as it will be discussed later.
The parameter Bn is going to be used in chapter §4 in order to be characterize the shape of the constellations resulting from the interaction between chromatic dispersion and fiber nonlinearities. In those cases we will see that the constellation shape is generally not symmetric over the two Cartesian quadratures.
In practice, in order to calculate the quantity Bn for a given signal, we first extract the signal samples corresponding to one particular symbol (for example, for QPSK signals, we initially extract one of the four possible QPSK states), we calculate Bn over the samples of this state and finally, we take the average of Bn over all states. For the calculation of Bn over each state, the quantities σΘ,estand σR,est are found in a straight-forward manner over the complex samples of the state, while Aest may also be directly found as the state complex average. Finally, σest may be calculated as the average of the standard deviations of the Cartesian coordinates σRe and σIm, i.e. σest = σRe+σIm
2 . However, we should underline that there exist different ways to calculate Aest and σest. For example, Aest may be alternatively calculated as the complex argument that maximizes the resulting
PDF, while σest may be estimated by the standard deviations following the axes of a rotated coordinate system, for example, with one of its axes passing from the state complex average. It is obvious that when the signal is degraded by AWGN, all the above variants converge into the same result. Nevertheless, we need to note that different choices of Aest and σest will generally result into slightly different estimations of the quantity Bn.
2.1.4.3.2 Calculation of the bit error probability for coherently
de-tected PSK
The probability of the M-ary symbol being detected correctly (noted as SCDPM) is equal to the probability of the detected phase lying inside the phase interval −Mπ , π M . Noting by pm = P 2m − 1 M π < Θ < 2m + 1 M π = 2m+1 M Z 2m−1 M π pΘ(θ)dθ (2.74)
SCDPM = p0 (2.75) Equivalently, the probability of the M -ary symbol being mistaken for another one, or the symbol error probability (noted as SEPM), is given by
SEPM = 1 − SCDPM = 1 − p0 (2.76)
Combining (2.57)and (2.76)we get
SEPM = 1 − e−ρ M − π M Z 0 !√ρ · e−ρ·sin2θ √ π cos (θ) erf c(− √ ρ · cos θ) " dθ (2.77)
The last integral of the equation(2.77)cannot be expressed in terms of existing functions for all arbitrary values of M and it generally has to be calculated numerically. However, for M = 2 an analytical solution exists and in that case the equation (2.77) reads
SEP2 = 1
2erf c(
√ρ) (2.78)
It is important to note that the equation (2.78) has the same form as the equation (2.47) and we may therefore conclude that the bit error probability is exactly the same for binaryASKandBPSK. This conclusion may also be reached with the simple observation made in section §2.1.3.2 that these two modulation formats have the exact same representation in the complex plane.
An analytical expression may also be provided for M = 4 by noting that the QPSK modulation is made-up from two independent BPSKsignals in phase quadrature. Since there is no interference between the two quadrature phase carriers and the noise on these carriers is statistically independent, the probability of a quaternary symbol being detected correctly is equal to the probability of both
BPSK symbols being correctly detected independently, or
SEP4 = erf c( r ρ 2) · 1 − 1 4ercf r ρ 2 (2.79)
1
3
5
7
9
11 13 15 17 1920
−6
−4
−2
0
ρ=SNR (dB)
log10(SEP)
BPSK
QPSK
8PSK
16PSK
32PSK
Figure 2.10: Symbol Error Probability as a function of SNR.
For the symbol error probabilities when M > 4 we use the formula (2.77), calculating the integral numerically.
Furthermore, in some cases, it is very useful to express symbol error proba-bilities as a function of the standard deviations of the polar coordinates, as, for example, in PSKmodulation, the information is coded in the phase of the optical signal. More specifically, as it will be discussed in more detail in chapter §4, we often use the standard deviation of the phase coordinate σΘas a rough estimation of the signal degradation and in this case, it is very useful to express the symbol error rates introduces above for PSK modulation, as a function of σΘ. While this is quite complicated in the general case, when ρ ≥ 10 we may combine the equations(2.71) and(2.78), to get an approximation of the symbol error rates in the case of BPSK SEP2 = 1 2erf c( 1 σΘ√2) (2.80) and QPSK SEP4 = erf c( 1 2σΘ) · 1 − 1 4ercf 1 2σΘ (2.81) , while these relations can be generalized to higher-level modulation formats.
Coming back to equations (2.78)-(2.77), in figure 2.10 we plot the resulting symbol error probability as a function of ρ for different values of M . As expected, higher level modulation requires higher levels ofSNRto achieve the same symbol error rate. However, this figure does not provide enough information on the “efficiency” of each modulation format. In other words, it doesn’t answer the ultimate question: “Which is the best modulation format for a transmission system limited by AWGN?”.
In order to answer this question we need to define what we mean by “efficient”. The useful and measurable qualities for a system are the bit error probability
BEP, the bit rate B, the bandwidth utilization W , the complexity, the cost, the throughput etc. Here we are going to neglect economical and implementation complexities and we are going to focus principally on the first three efficiency criteria.
At first, we need to convert the symbol error rate, for the curves of figure
2.10, to an equivalent bit error rate for all modulation formats. For this, we need the information of how the different bits are mapped into symbols, since a wrong decision over a symbol may not necessarily imply a wrong decision over all log2M bits of the symbol. In practice, it is more probable that, due to noise, a symbol error will most possibly signify mistaking the symbol for one of its closest neighbors on the complex plane. This information can be exploited by designing the mapping in such a way that symbols with an adjacent phase (in the case of
PSKsignals) represent tuples that differ to only 1 bit. It can be shown that this is the optimum mapping, also known as Gray encoding[89]. An accurate evaluation
of the BEP in the case of Gray coded MPSK was presented in [70]. However, a fairly accurate simple approximation may be considered by noting that in the case of Gray encoding, a symbol error will yield in most times a single bit error. In this case, the bit error probability is given as a function of the symbol error probability by the formula
BEPM = SEPM
log2M (2.82)
Secondly, we need to note that different modulation formats (and therefore the different curves of figure 2.10) naturally provide a different bit rate B for the
same amount of bandwidth W , as each symbol carries a tuple of log2M bits. The symbol rate R is linked to the bit rate BM (the index M signifies the symbol rate for M -ary PSK) by the formula
B = RM · log2M (2.83)
Furthermore, for the purposes of this analysis we admit that this bandwidth occupied by the signal is equal to the symbol rate R1, i.e.
WM=RM= B
log2M = W2
log2M (2.84)
and modulation of order M > 2 occupies log2M less bandwidth than modu-lation with M = 2.
From the above it is obvious that a simple criterion like the “SNR required to deliver a certain bit error probability” is not sufficient to describe the quality of a signal. For this we need to introduce a new quantity that takes the SNR
requirement down to the level of one bit. The SN R/bit is defined by replacing W in the equation (2.29) by the bitrate B as defined in the equation (2.83)
ρb = P
N0B
(2.85) or combining the equations(2.85) and (2.84)
ρb = ρ
log2M (2.86)
In figure 2.11 we plot the BEP versus ρb. An interesting remark concerning this figure is thatBPSKandQPSKhave the exact same performance in terms of
BEP as a function of ρb. However, the notable difference between the solutions
BPSK and QPSK is that QPSK naturally uses half the bandwidth than BPSK.
1There exist several definitions for the notion of bandwidth (see[102]). In optical commu-nications, the definition mostly used is the null-to-null bandwidth, i.e. if the symbol shaping pulse used has the form of rect t
T
with T = R−1 being the symbol period, then its Fourier transform is T · sinc (T · t) a function with its central lobe-and most of the spectrum contained into T2 = 2R. In practice, when tight optical filtering is applied at the reception or for channel extraction, its bandwidth is about 2R, or a little less. The difference with the assumption of this analysis is just a factor of 2.
−5
0
5
10
15
20
−6
−4
−2
0
ρ/bit=SNR/bit (dB)
log10(BEP)
BPSK
8−PSK
16−psk
QPSK
Figure 2.11: BEP Vs SNR/bit.
Therefore QPSK and BPSK have the same performance in terms of BEP, but
somewhat,QPSKuses more efficiently the given spectrum. In order to distinguish
BPSK and QPSK with respect to this last “quality”, we usually refer to the quantity “spectral efficiency”1 defined as
η=∆ B
W (2.87)
In order to take η into account in the comparison of different modulation formats, we need to normalize the amount of bandwidth used by each modulation format, or equivalently, compare the different solutions by fixing the amount of bandwidth used. To simplify the analysis, without loss of generality, we may consider that for all values of M the signal power is fixed and the noise spectral density is also fixed. Consequently, SNR depends only on the bandwidth WM occupied by an MPSK signal.
1In the context of dense Wavelength-Division Multiplexing (WDM) optical communication systems (which is the case for this work) spectral efficiency of a given modulation format is of extreme importance.
0
2
4
6
8
−6
−4
−2
0
SNR/(bit
2
) (dB)
log10(BEP)
BPSK
QPSK
8−PSK
16−PSK
Figure 2.12: BEP Vs SN R/bit2.
Normalizing the formats to the same bandwidth utilization (or the same spec-tral efficiency), we define the quantity SN R/bit2 as
ρb2 = ρ
(log2M )2 (2.88)
In figure 2.12 we plot the BEP as a function of ρb2. It is evident that QPSK
provides the best solution with respect to this combined criterion of required SN R/bit and spectral efficiency, whereas 8-PSK, BPSK and 16-PSK follow.
Optical telecommunications were dominated for a long time by classicalOOK
modulated signals. As we have seen in section §2.1.4.2, the BEP of OOK -modulated signals impacted by AWGN can be expressed in terms of a quantity that called the Q factor by a “1-1” relation (eq. (2.47)). Nevertheless, in opti-cal communications, even when it comes to non-OOK modulation, the measured
BERs is very often converted into a “fake Q factor” by inverting eq. (2.47). Apart from reasons of comparison and “backward compatibility”, this conversion also offers the advantage that when it comes to binary modulated signals impacted byAWGN, Q2 is a linear function of theSNRand the signal quality for any SNR
